On the Space of Slow Growing Weak Jacobi Forms

In this note we investigate the asymptotic behavior of the Fourier coefficients of certain types of automorphic forms, namely weak Jacobi forms [EZ85] and their exponential lifts, which give meromorphic Siegel (para-)modular forms [GN98]. The growth of coefficients of automorphic forms has of course long been of interest in mathematics: the most famous example may be the work of Hardy and Ramanujan, who found asymptotic expressions for the number of integer partitions from the Dedekind $\eta$-function [HR00]. However, more recently, the growth of coefficients of automorphic forms has also attracted interest in physics: it describes the entropy of certain types black holes [SV96, Sen11]. This was explored from a more mathematical perspective for instance in [DMZ12]. The main motivation for this works comes from a series of articles [BCKM19, BCKM20, BBC⁺20] that were written in this context. There it was discovered that weak Jacobi forms could either lead to fast growth or slow growth in their exponential lifts. Our goal is to investigate the space of slow growing forms.

Let us introduce the main definition of this article. Let $\varphi$ be a weak Jacobi form (wJf) [EZ85] of weight 0 and index $m$ with Fourier expansion

Given a term $q^{n}y^{l}$ in this expansion, we call the quantity $l^{2}-4mn$ its polarity, and we say that the term is polar if its polarity is positive. Let $q^{a}y^{b}$ be such a polar term of $\varphi$. We want to study the functions

and their asymptotic growth as $n$ becomes large. Note that this sum is a finite sum because $c(n,l)=0$ if $l^{2}-4mn>m^{2}$. We will explain the motivation behind this definition below. We make the following definition:

The purpose of this article is to describe and explore the spaces $\mathcal{J}^{a,b}_{m}$. Our ultimate goal is to prove the following conjecture:

To explain our interest in this conjecture, note that it is somewhat surprising that slow growing forms exist in the first place: If a wJf has a term of maximal polarity $q^{a}y^{b}$, then its coefficients grow like

in the limit of large discriminant $4mn-l^{2}\gg 0$. For (2) to be bounded, the exponentially large terms thus have to cancel out almost completely, which is not something we would expect for a generic form $\varphi$. However, there are indeed examples where such cancellations happen. The simplest is $\varphi_{0,1}$, the unique (up to normalization) wJf of weight 0 and index 1. In this note we provide evidence that such slow growing forms exist for every index $m$. Even though we do not manage to prove this conjecture, we gather numerical and analytical evidence in favor of it.

Let us now explain where definition (2) comes from, and why we are interested in slow growing wJf in the first place. A wJf of weight 0 can be exponentially lifted to a meromorphic Siegel paramodular form [GN98]. It is the Fourier coefficients of this Siegel form that we want to study. Since this form meromorphic, we need to be careful to specify which region we are expanding in when defining its Fourier coefficients. Its poles are given by Humbert surfaces specified by $a,b$. $f_{a,b}(n,l)$ then describes the growth of the Fourier coefficients in an appropriate limit when expanded around that particular Humbert surface [Sen11, BCKM19]. The most basic example of such a lift is the wJf $\varphi_{0,1}$, which is lifted to the reciprocal of the Igusa cusp form $\chi_{10}$. This is the case originally studied by physicists to describe the entropy of certain black holes [SV96], and was further explored in [DMZ12]. Our results are a generalization of this case.

We note that for the purpose of obtaining Fourier coefficients of SMF, strictly speaking we are not interested in $\mathcal{J}^{a,b}_{m}$, but rather in the subset

A form in $\hat{\mathcal{J}}^{a,b}_{m}$ is guaranteed to lead to a (simple) pole in the corresponding exponentially lifted Siegel form. $\hat{\mathcal{J}}$ is an affine space. If it is non-empty, that is if $\mathcal{J}$ contains at least one form with $c(a,b)=1$, then $\dim\hat{\mathcal{J}}=\dim\mathcal{J}-1$, since the linear constraint $c(a,b)=0$ on $\mathcal{J}$ has rank 1. It turns out this is almost always what happens. In the following we will therefore always give $\dim\mathcal{J}$, and indicate explicitly the few cases when $\hat{\mathcal{J}}=\emptyset$.

The Fourier coefficients of Siegel modular forms give one way in which the $f_{a,b}(n,l)$ are connected to automorphic forms. Let us mention that for $a=0$, there is also another connection: in [BCKM19] it was shown that the $f_{0,b}(n,l)$ are Fourier coefficients of certain vector valued modular functions. In fact this gave an explicit test for when a wJf is slow growing, and very simple expressions for the $f_{0,b}(n,l)$ in this case — see theorem 6 below. There is no analogue test for $a>0$, and the situation is thus much less clear. We discuss the case $a>0$ in section 4 and find some evidence that the $f_{a,b}$ could again be coefficients of some modular object.

Finally let us point out that slow growth seems to be related to the maximal polarity of the terms appearing in a form. In [BCKM19] for instance it was established that $\varphi$ can only be slow growing about $q^{0}y^{b}$ if $b^{2}\leq m$. In section 4 we give examples of forms growing slowly about $q^{a}y^{b}$ whose maximal polarity is $b^{2}-4ma>m$, but only slightly so. The implication that slow growing forms can only have terms of relatively low polarity. It thus makes sense to look for forms of low polarity. A priori it is not clear that such wJf even exist for any $m$, regardless of any question of slow growth: The issue is that even though for weight 0 forms, the polar terms uniquely determine $\varphi$, it is not guaranteed that for a choice of polar terms there exists a corresponding wJf $\varphi$. It is believed [GGK⁺08] that for any $m$, there exist wJfs whose most polar term has polarity around $m/2$. In particular this allows for the existence of slow growing wJf. In section 2 we provide strong numerical evidence for this belief, and in particular establish the existence of such forms up to $m=1000$. We note that a similar question is discussed in [DMZ12], where so-called optimal wJf ${\mathcal{K}}_{m}\in J_{2,m}$ are constructed, whose maximal polarity is 1. It is natural to believe that $\varphi_{-2,1}{\mathcal{K}}_{m}$ is then a slow growing form for index $m+1$, which is indeed true for the cases that we checked. A proof of this belief would of course prove conjecture 1.

In section 3 we then discuss weak Jacobi forms that have slow growth around $q^{0}y^{b}$. Following up on an observation in [BBC⁺20], we give infinite families of weak Jacobi forms based on quotients of theta functions that are all slow growing. We obtain bounds on the $\dim\mathcal{J}^{0,b}_{m}$, and compute these dimensions explicitly up to $m=61$. Altogether we establish that slow growing forms exist for every index up to $m=78$.

In section 4 we give some analytical and numerical results on the growth about terms $q^{a}y^{b}$. This case is harder because unlike $a=0$, there is no longer a straightforward criterion to determine if a form is slow growing. Nonetheless, we manage to prove slow growth for certain cases. In the process we clarify a question raised in [BCGK18], where an exponentially lifted Siegel modular form with two Humbert surface of the same discriminant was considered. The physical expectation was that the expansion around both surfaces should essentially give the same coefficients, which is what we prove here. For many other cases we give strong numerical evidence for or against slow growth by evaluating the first few coefficients, giving a better idea on the form of $\mathcal{J}^{a,b}_{m}$. We give examples of forms growing slowly about $q^{a}y^{b}$ whose maximal polarity is $b^{2}-4ma>m$, and we also give examples that are slow growing about one term but fast growing about another term of the same polarity, thus establishing that the polarity of the term is not enough to determine the type of growth.

Acknowledgments: We thank Nathan Benjamin for useful discussions. The work of C.A.K. is supported in part by the Simons Foundation Grant No. 629215.

Let $J_{0,m}$ be the space of weak Jacobi forms of weight $0$ and index $m$ [EZ85]. Its structure is very simple: the ring of wJf of weight 0 is freely generated by three forms $\phi_{0,1}$, $\phi_{0,2}$, $\phi_{0,3}$ given e.g. in [Gri99]. $J_{0,m}$ is then spanned by products $\phi_{0,1}^{a}\phi_{0,2}^{b}\phi_{0,3}^{c}$, with $a+2b+3c=m$.

In practice however, working with wJf of already moderate index can be quite cumbersome, since it involves expanding products of series expansions to fairly high order. It is therefore important to have computationally efficient expressions for the generators. We used

and

We are using this novel form because it can be evaluated more quickly, as it requires only few multiplications, and no divisions by expressions in multiple variables.

As mentioned in the introduction, wJf cannot have slow growth about $q^{0}y^{b}$ if $b>\sqrt{m}$. We therefore want to investigate wJf with no terms of high polarity. To this end we define

$P(m)$ is well-defined since $J^{P}_{0,m}\subset J^{P+1}_{0,m}$, $J^{0}_{0,m}=0$ for $m>0$, and moreover $J^{m^{2}}_{0,m}=J_{0,m}$. In particular definition 2 implies that there is a non-zero wJf whose most polar term has polarity $l^{2}-4mn=P(m)$, but no no-zero wJf that only has terms of polarity strictly smaller than $P(m)$.

Given the generators of $J_{0,m}$, in principle it is straightforward to compute $P(m)$. In practice this involves expanding power series to high order to read off polar terms, which becomes computationally expensive very quickly. Using (5-7), we computed $P(m)$ to index $m=61$. The result is plotted in figure 1.

We are interested in the asymptotic behavior of $P(m)$, which unfortunately is hard to compute directly. For large $m$, [GGK⁺08] conjectured the following form for $P(m)$:

In what follows, we will give upper and lower bounds $P_{+}(m)\geq P(m)\geq P_{-}(m)$ for $P(m)$. The bounds we find are indeed compatible with conjecture 2. Let us first discuss the lower bound $P_{-}(m)$ for $P(m)$. To this end, we prove the following proposition, which strengthens a result in [Man08]:

From this it follows immediately that

Next let us discuss upper bounds $P_{+}(m)$. One way to obtain such a bound is the following result of [DMZ12]:

From this we obtain the following corollary:

It follows that $P_{+}(m)=m+1$ is an upper bound for $P(m)$. Unfortunately, this bound is slightly too weak for our purposes, since we would like to find wJf of polarity $m$ or less. Nonetheless, we note that the forms obtain in this way are natural candidates for being slow growing. In fact, for the first few low lying values of $m$ we checked explicitly that they are slow growing. This is the case even for those with maximal polarity equal to $m+1$, in which case the term of maximal polarity is $q^{a}y^{b}$ with $a>0$.

More importantly, the bound $P_{+}(m)=m+1$ also seems to be quite far from optimal when compared to conjecture 2. Let us therefore describe an alternative approach, which will lead to a tighter numerical bound. For this we use the following counting argument:

Since $P^{+}(m)$ only involves counting polar terms, it is very easy to compute. We present a scatter plot of its value for $1\leq m\leq 1000$ in Figure 2.

We expect that for generic $m$, $P(m)$ will be very close to $P^{+}(m)$. Equality between $P(m)$ and $P^{+}(m)$ holds whenever the linear system of polar coefficients with polarity greater than or equal to $P^{+}(m)$ has maximal rank. We expect the matrix of these polar coefficients to behave like a random matrix, and such matrices generically have maximal rank. Comparing Figure 2 with the scatter plot for $P(m)$ for $1\leq m\leq 61$ in Figure 1, we find that $P(m)=P^{+}(m)$ except at $m=39,51,54$, and $58$. A particularly wide gap is found at $m=54$, where $P(m)=9$ but $P^{+}(m)=25$. In contrast to $P^{+}(m)$, we expect $P^{-}(m)$ to be a weak lower bound. Numerically, up to $m=1000$ we find that $|P^{+}(m)-\frac{m}{2}|\leq 2.1016m^{1/2}$.